Complex Dielectric Constant Extraction of Substrate Materials Using Cross-Resonator Method

Abstract

This article presents a novel resonance method by using a cross-resonator (CR) for the complex permittivity measurement of planar dielectric substrate materials with its full theoretical and analytical transmission line (TL) model. Due to its symmetrical structure, it offers two main advantages over its conventional counterpart. First, the unwanted asymmetricity mode is eliminated due to the centric magnetic wall. Second, the electric field at the junction of the open-ended stub is uniform. The propagation constant of the structure in this method is directly extracted from the <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>-parameters. The total losses without approximation are computed by using a robust theoretical and analytical TL modeling. The proposed resonator is implemented on RO4003C and FR4 substrate materials, and <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>-parameters are measured in the frequency range of 1 MHz to 8 GHz. Then the complex dielectric constant of the substrate is extracted at each resonant frequency. A full-wave simulation also confirms the accuracy of the proposed resonator. The comparison between the simulated and the measured <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>-parameters demonstrate a relative error less than 1&#x0025; over the frequency range.